Introducing the definitions of two kinds of inner product and utilizing the principle of mechanical energy conservation of a conservative system, this paper presents the general form of orthogonality relations and unifiedproof for the system to be considered.

On the basis of Hamilton's principle and a linear potential flow theory,generalized orthogonality relations of wet modes for 3—D fluid-structure system are proved. The decoupling conditions of the equations of motion are also discussed.

A exponential generating function was introduced by using of the method of combinatorial analysis; Two new analysis expressions about stirling number of the second kind were given through iteration by using of orthogonality relations.

The identified modal matrix is decomposed by means of singular-value decomposition technique combining with the eigenequation and modal orthogonality relations, and thus general modified equation for the analytical model is obtained.

Using orthogonality relations of Fourier series,we reduce governing nonlinear partial differential equations to an infinite set of system of nonlinear algebraic equations containing Fourier cofficients.

The mode shapes can satisfy all the boundary conditions of plate and possess the orthogonality relations between themselves. The forms of the mode shapes are accordant with the deformation patterns permitted by the edge supports.

Half-range orthogonality relations basic to the solution of time-dependent boundary value problems in the kinetic theory of gase

Characterizations of real inner product spaces among normed linear spaces have been obtained by exploring properties of and relationships between various orthogonality relations which can be defined in such spaces.

We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials.

In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.

It is based on the orthogonality relations for uniform B-splines in weighted Sobolev spaces, as introduced in (Reif, 1997).

Tellegen's theorem in various forms is derived by the orthogonality relation between the current subspace and voltage subspace. It is shown that the physical meaning of Tellegen's theorem can be taken as an expression of the law of energy conservation for a network in different states, and it is understood that a network state can be specified simultaneously with any vector j in current subspace and any vector V in voltage subspace.

Introducing the definitions of two kinds of inner product and utilizing the principle of mechanical energy conservation of a conservative system, this paper presents the general form of orthogonality relations and unifiedproof for the system to be considered. At the end of this paper, some applications to various structures are presented.

In this paper, according to the characteristics of linear vibrating defective systems, a complete set of linearly independent generalized eigenvectors is found for the defective system. Such eigenvectors many be called as generalized modes. The partially bi-orthogonality relation between the generalized modes of the original and the adjoint systems i's also found and demonstrated. By use of the bi-orthogonality relation, the most general expressions of the transfer function and the pulse response...

In this paper, according to the characteristics of linear vibrating defective systems, a complete set of linearly independent generalized eigenvectors is found for the defective system. Such eigenvectors many be called as generalized modes. The partially bi-orthogonality relation between the generalized modes of the original and the adjoint systems i's also found and demonstrated. By use of the bi-orthogonality relation, the most general expressions of the transfer function and the pulse response function of linear vibrating system are derived.A sample example is given to show the correctness and effectiveness of the suggested theory and method.